Augmented matrices with at least one solution For what value

Augmented matrices with at least one solution

For what value(s) of h do the augmented matrices below represent systems of linear equations with at least one solution? [1 h -3 -2 4 6] [2 -3 h -6 14 5]

Solution

rewriting in equatoion form

1x + hy =-3

-2x +8y = 6

multiplying 1st equation by 2

2x +2hy = -6

-2x +8y = 6

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adding both

(2h+8) y =0

then 2h+8 =0

=>h =-4

or y =0

2nd equation

2x -3y =h

-6x+14y = 5

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multiply 1st equation by 3 and adding

6x -9y = 3h

-6x+14y = 5

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0 +5y = 3h+5

3h = 5(y-1)

h = 5(y-1)/3

no conclusion can be made about value of h